Let be $J$ a convex functional defined in Hilbert space H and with real values. What hypothesis I should assume to exist solution for the problem?:

$J(u) = \inf \left\{{J(v); v \in K}\right\} , u \in K$ For all closed convex $K \subset H.$

I begin using the theorem

A functional $J:E\rightarrow\mathbb{R}$ defined over a norm space $E$ is semi-continuous inferiorly if for all sequence $(u_n)_{n\in \mathbb{N}}$ converging to $u$ then: $\lim_{n\rightarrow \infty}\inf J(u_n)\geq J(u)$.

But I don't know how make to only "=".

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You get equality by taking $u_n$ such that $J(u_n)\to \inf_K J$. Indeed, the weak limit is also an element of $K$ and therefore cannot have a smaller value of the functional than the infimum.
The term is "lower semicontinuous", by the way. What you need from $J$ is being bounded from below, and lower semicontinuous with respect to weak convergence of sequences. And if you allow unbounded $K$, it helps to have $J\to\infty$ at infinity, because this forces the sequence $u_n$ to be bounded.